Toffoli gate preparation for a quantum hardware system comprising hybrid acoustic-electrical qubits

Abstract

A Toffoli magic state to be injected in preparation of a Toffoli gate may be prepared using a bottom-up approach. In the bottom-up approach, computational basis states are prepared in a fault tolerant manner using a STOP algorithm. The computational basis states are further used to prepare the Toffoli magic state. The STOP algorithm tracks syndrome outcomes and can be used to determine when to stop repeating syndrome measurements such that faults are guaranteed to be below a threshold level. Also, the STOP algorithm may be used in growing repetition code from a first code distance to a second code distance, such as for use in the computational basis states.

What is claimed is: 
     
       1. A method, comprising:
 measuring syndrome outcomes of an ancilla qubit for an arbitrary Calderbank-Shor Steane code; 
 tracking consecutive ones of the measured syndrome outcomes; 
 computing a minimum number of faults capable of causing a tracked sequence of consecutive syndrome outcomes; 
 stopping the measuring of the syndrome outcomes if either of the following conditions is met:
 1) a same syndrome outcome is repeated a threshold number of times in a row, wherein the threshold is equal to one plus a difference between:
 a code distance of the arbitrary Calderbank-Shor-Steane code minus one, divided by two; and 
 a currently computed minimum number of faults capable of causing the tracked sequence of consecutive syndrome outcomes; or 
 
 2) the currently computed minimum number of faults capable of causing the tracked sequence of consecutive syndromes is equal to the code distance minus one, divided by two, and wherein one additional round of syndrome measurements is performed subsequently; and 
 
 utilizing the repeated syndrome outcome if condition 1 is met or utilizing the syndrome outcome for the subsequently performed syndrome measurement if condition 2 is met, wherein the utilized syndrome outcome is utilized to error correct the arbitrary Calderbank-Shor-Steane code. 
 
     
     
       2. The method of  claim 1 , wherein:
 the arbitrary Calderbank-Shor-Steane code is a n-qubit repetition code; 
 measuring the syndrome outcomes comprises measuring Z L  at the ancilla for the n-qubit repetition code; and 
 performing the error correction for the n-qubit arbitrary Calderbank-Shor-Steane code further comprises applying an X L  correction based on the measured Z L  at the ancilla for the n-qubit repetition code, 
 wherein performing the error correction prepares computational basis states to be used in implementing a Clifford gate. 
 
     
     
       3. The method of  claim 1 , further comprising:
 growing the arbitrary Calderbank-Shor-Steane code from a first code distance to a second code distance, wherein a STOP algorithm is used to measure stabilizers and minimum weight perfect matching (MWPM) is applied to a measured syndrome history generated from measuring the stabilizers to correct for errors, wherein the STOP algorithm comprises said measuring syndrome outcomes, said tracking consecutive ones of the measured outcomes, said computing a minimum number of faults, said stopping the measuring if condition 1 or condition 2 is met, and said error correction. 
 
     
     
       4. The method of  claim 3 , wherein said growing the arbitrary Calderbank-Shor-Steane code from the first code distance to the second code distance comprises:
 performing lattice surgery to merge together two code blocks, wherein the measuring comprises measuring a boundary operator between the two code blocks being merged. 
 
     
     
       5. The method of  claim 1 , further comprising:
 preparing computational basis states in a fault tolerant manner by applying a STOP algorithm to determine when syndrome measurements of stabilizers of a repetition code for the computational basis states can be stopped such that a probability of faults for the computational basis states are below a threshold level, 
 wherein:
 the computational basis states are encoded using the arbitrary Calderbank-Shor-Steane code; and 
 applying the STOP algorithm comprises performing said measuring syndrome outcomes, said tracking consecutive ones of the measured outcomes, said computing a minimum number of faults, said stopping the measuring if condition 1 or condition 2 is met, and said error correction. 
 
 
     
     
       6. The method of  claim 5 , further comprising:
 transversally applying a CNOT gate to the prepared computational basis states to prepare a |ψ 1    state; 
 measuring a Clifford stabilizer g A  for the |ψ 1    state, and applying a logical Z correction if the measurement outcome for the Clifford stabilizer g A  is −1, wherein measuring the Clifford stabilizer g A  and applying the logical Z correction based on a measurement outcome of the Clifford stabilizer g A  prepares a state |ψ out   ; 
 repeating the Clifford stabilizer g A  measurement for the |ψ 1    state a threshold number of times; 
 preparing a Toffoli magic state in response to determining the Clifford stabilizer g A  measurements are trivial; and 
 applying a sequence of Clifford gates to the |ψ 1    state and the prepared Toffoli magic state to simulate a Toffoli gate, wherein Clifford error corrections are applied to the outputs of the sequence of Clifford gates applied to a logical input. 
 
     
     
       7. The method of  claim 6 , wherein:
 repeating the measurement of the Clifford stabilizer g A  for the |ψ 1    state the threshold number of times comprises repeating the measurement such that the Clifford stabilizer g A  is measured a number of times equal to (d−1)/2, wherein d is a code distance of the one of the fault tolerant computational basis states. 
 
     
     
       8. The method of  claim 7 , wherein error detection is performed between respective measurements of the Clifford stabilizer g A . 
     
     
       9. The method of  claim 8 , further comprising:
 growing the Toffoli magic state from a first code distance to a second code distance, wherein the STOP algorithm is used to measure stabilizers and minimum weight perfect matching (MWPM) is applied to a measured syndrome history generated from measuring the stabilizers to correct for errors. 
 
     
     
       10. The method of  claim 1 , wherein the arbitrary Calderbank-Shor-Steane code and the ancilla are implemented using a system comprising:
 mechanical linear resonators; and 
 a control circuit coupled with the mechanical linear resonators, 
 wherein the control circuit is configured to stabilize an arbitrary coherent state superposition (cat state) of the mechanical linear resonators to store quantum information of the Calderbank-Shor-Steane code, wherein to stabilize the arbitrary cat-state, the control circuit is configured to:
 excited phonons in the mechanical linear resonators by driving respective storage modes of the mechanical linear resonators; and 
 dissipate phonons from the mechanical linear resonators via an open transmission line coupled to the control circuit configured to absorb photons from a dump mode of the control circuit. 
 
 
     
     
       11. The method of  claim 10 , wherein the control circuit comprises:
 an asymmetrically-threaded superconducting quantum interference device (ATS) coupled with the mechanical linear resonators. 
 
     
     
       12. A system, comprising:
 mechanical resonators; and 
 a control circuit coupled with the mechanical resonators, wherein the control circuit is configured to stabilize arbitrary coherent state superpositions (cat states) of the mechanical resonators to store quantum information; and 
 one or more computing devices storing program instructions, that when executed cause the control circuit to perform:
 measuring syndrome outcomes of an ancilla qubit for one or more qubits storing the quantum information, wherein the ancilla qubit and the one or more qubits storing the quantum information are implemented via one or more of the mechanical resonators; 
 tracking consecutive ones of the measured syndrome outcomes; 
 computing a minimum number of faults capable of causing a tracked sequence of consecutive syndrome outcomes; 
 stopping the measuring of the syndrome outcomes if either of the following conditions is met:
 1) a same syndrome outcome is repeated a threshold number of times in a row, wherein the threshold is equal to one plus a difference between:
 a code distance of the one or more qubits storing quantum information minus one, divided by two; and 
 a currently computed minimum number of faults capable of causing the tracked sequence of consecutive syndrome outcomes; or 
 
 2) the currently computed minimum number of faults capable of causing the tracked sequence of consecutive syndromes is equal to the code distance minus one, divided by two, and wherein one additional round of syndrome measurements is performed subsequently; and 
 
 utilizing the repeated syndrome if condition 1 is met or utilizing the syndrome outcome for the subsequently performed syndrome measurement if condition 2 is met, wherein the utilized syndrome outcome is utilized to error correct the stored quantum information. 
 
 
     
     
       13. The system of  claim 12 , wherein the one or more computing devices are further configured to implement:
 preparing computational basis states in a fault tolerant manner by applying a STOP algorithm to the fault-tolerant computational basis states to determine when syndrome measurements of stabilizers of a repetition code for the computational basis states can be stopped such that a probability of faults for the computational basis states are below a threshold level, 
 wherein:
 applying the STOP algorithm comprises performing said measuring syndrome outcomes, said tracking consecutive ones of the measure outcomes, said computing a minimum number of faults, said stopping the measuring if condition 1 or condition 2 is met, and said error correction. 
 
 
     
     
       14. The system of  claim 12 , wherein the one or more computing devices are further configured to implement:
 transversally applying a CNOT gate to the prepared computational basis states to prepare a |ψ 1    state; 
 measuring a Clifford stabilizer g A  for the |ψ 1    state, and applying a logical Z correction if the measurement outcome for the Clifford stabilizer g A  is −1, wherein measuring the Clifford stabilizer g A  and applying the logical Z correction based on a measurement outcome of the Clifford stabilizer g A  prepares a state |ψ out   ; 
 repeating the Clifford stabilizer g A  measurement for the |ψ 1    state a threshold number of times; 
 preparing a Toffoli magic state in response to determining the Clifford stabilizer g A  measurements are trivial; and 
 applying a sequence of Clifford gates to a logical input state |ψ   L  and the prepared Toffoli magic state to simulate the Toffoli gate, wherein Clifford error corrections are applied to the outputs of the sequence of Clifford gates applied to the logical inputs. 
 
     
     
       15. The system of  claim 14 , wherein the one or more computing devices are further configured to implement:
 growing the Toffoli magic state from a first code distance to a second code distance, wherein the STOP algorithm is used to measure stabilizers and minimum weight perfect matching (MWPM) is applied to a measured syndrome history generated from measuring the stabilizers to correct for errors. 
 
     
     
       16. One or more non-transitory computer-readable media storing program instructions, that when executed on or across one or more processors, cause the one or more processors to:
 receive syndrome measurement outcomes of an ancilla qubit for an arbitrary Calderbank-Shor Steane code; 
 track consecutive ones of the measured syndrome outcomes; 
 compute a minimum number of faults capable of causing a tracked sequence of consecutive syndrome outcomes; 
 stop performing of the syndrome measurements if either of the following conditions is met:
 1) a same syndrome measurement outcome is repeated a threshold number of times in a row, wherein the threshold is equal to one plus a difference between:
 a code distance of the arbitrary Calderbank-Shor-Steane code minus one, divided by two; and 
 a currently computed minimum number of faults capable of causing the tracked sequence of consecutive syndrome measurement outcomes; or 
 
 2) the currently computed minimum number of faults capable of causing the tracked sequence of consecutive syndromes is equal to the code distance minus one, divided by two, and wherein one additional round of syndrome measurements is performed subsequently; and 
 
 utilizing the repeated syndrome measurement outcome if condition 1 is met or utilizing the syndrome measurement outcome for the subsequently performed syndrome measurement if condition 2 is met, wherein the utilized syndrome measurement outcome is utilized to error correct the arbitrary Calderbank-Shor-Steane code. 
 
     
     
       17. The one or more non-transitory, computer-readable storage media of  claim 16 , wherein the program instructions, when executed on or across the one or more processors cause the one or more processors to:
 cause the arbitrary Calderbank-Shor-Steane code to be grown from a first code distance to a second code distance, 
 wherein a STOP algorithm is used to measure stabilizers and minimum weight perfect matching (MWPM) is applied to a measured syndrome history generated from measuring the stabilizers to correct for errors, 
 wherein the STOP algorithm comprises measuring the received syndrome measurement outcomes, said tracking consecutive ones of the measured outcomes, said computing a minimum number of faults, said stopping the measuring if condition 1 or condition 2 is met, and said error correction. 
 
     
     
       18. The one or more non-transitory, computer-readable media of  claim 16 , wherein to grow the arbitrary Calderbank-Shor-Steane code from the first code distance to the second code distance the program instructions cause:
 lattice surgery to be performed to merge together two code blocks, wherein the measuring comprises measuring a boundary operator between the two code blocks being merged. 
 
     
     
       19. The one or more non-transitory, computer-readable storage media of  claim 16 , wherein the program instructions, when executed on or across the one or more processors cause the one or more processors to:
 cause computational basis states to be prepared in a fault tolerant manner by applying a STOP algorithm to determine when syndrome measurements of stabilizers of a repetition code for the computational basis states can be stopped such that a probability of faults for the computational basis states are below a threshold level, 
 wherein:
 the computational basis states are encoded using the arbitrary Calderbank-Shor-Steane code; and 
 applying the STOP algorithm comprises measuring the received syndrome measurement outcomes and performing said tracking consecutive ones of the measured outcomes, said computing a minimum number of faults, said stopping the measuring if condition 1 or condition 2 is met, and said error correction. 
 
 
     
     
       20. The one or more non-transitory, computer-readable storage media of  claim 19 , wherein the program instructions, when executed on or across the one or more processors cause the one or more processors to:
 cause a CNOT gate to be transversally applied to the prepared computational basis states to prepare a |ψ 1    state; 
 cause a Clifford stabilizer g A  to be measured for the |ψ 1    state, and cause a logical Z correction to be applied if the measurement outcome for the Clifford stabilizer g A  is −1, wherein measuring the Clifford stabilizer g A  and applying the logical Z correction based on a measurement outcome of the Clifford stabilizer g A  prepares a state |ψ out   ; 
 cause the Clifford stabilizer g A  measurement to be repeated for the |ψ 1    state a threshold number of times; 
 cause a Toffoli magic state to be prepared in response to determining the Clifford stabilizer g A  measurements are trivial; and 
 cause a sequence of Clifford gates to be applied to the |ψ 1    state and the prepared Toffoli magic state to simulate a Toffoli gate, wherein Clifford error corrections are applied to the outputs of the sequence of Clifford gates applied to a logical input.